Required Prior Knowledge

Questions

Label the size of the angles in these two triangles in both degrees and radians.

Solutions

Get Ready

Questions

Using the two triangles above and the unit circle, fill in the table below.

Solutions

Your Turn

Without looking at the table above, fill in the gaps so that all cells in a row have the same value.

Examples and Your Turns

Example

Find the exact values of $\sin\alpha$, $\cos\alpha$ and $\tan\alpha$ for $\alpha =\frac{3\pi}{4}$.

Your Turn

Find the exact values of $\sin\alpha$, $\cos\alpha$ and $\tan\alpha$ for $\alpha =\frac{4\pi}{3}$.

Your Turn

Find the exact values of $\sin\alpha$, $\cos\alpha$ and $\tan\alpha$ for $\alpha =225^{\circ}$.

Your Turn

Find the exact values of $\sin\alpha$, $\cos\alpha$ and $\tan\alpha$ for $\alpha =\frac{5\pi}{3}$.

Your Turn

Find the exact values of $\sin\alpha$, $\cos\alpha$ and $\tan\alpha$ for $\alpha =\frac{11\pi}{6}$.

Your Turn

Find the exact values of $\sin\alpha$, $\cos\alpha$ and $\tan\alpha$ for $\alpha =510^{\circ}$.

Your Turn

Find the exact value for each of these.

  
          a) \(\sin 210^{\circ}\)

                $$\begin{align*} \sin 210^{\circ} &= -\sin 30^{\circ} \\ &= -\frac{1}{2} \end{align*}$$
              ?

          b) \(\sin 150^{\circ}\)

                $$\begin{align*} \sin 150^{\circ} &= \sin 30^{\circ} \\ &= \frac{1}{2} \end{align*}$$
              ?

          c) \(\sin 225^{\circ}\)

                $$\begin{align*} \sin 225^{\circ} &= -\sin 45^{\circ} \\ &= -\frac{\sqrt{2}}{2} \end{align*}$$
              ?

          d) \(\cos 300^{\circ}\)

                $$\begin{align*} \cos 300^{\circ} &= \cos 60^{\circ} \\ &= \frac{1}{2} \end{align*}$$
              ?

          e) \(\cos 150^{\circ}\)

                $$\begin{align*} \cos 150^{\circ} &= -\cos 30^{\circ} \\ &= -\frac{\sqrt{3}}{2} \end{align*}$$
              ?

          f) \(\cos 135^{\circ}\)

                $$\begin{align*} \cos 135^{\circ} &= -\cos 45^{\circ} \\ &= -\frac{\sqrt{2}}{2} \end{align*}$$
              ?

          g) \(\tan 240^{\circ}\)

                $$\begin{align*} \tan 240^{\circ} &= \tan 60^{\circ} \\ &= \sqrt{3} \end{align*}$$
              ?

          h) \(\tan 150^{\circ}\)

                $$\begin{align*} \tan 150^{\circ} &= -\tan 30^{\circ} \\ &= -\frac{\sqrt{3}}{3} \end{align*}$$
              ?

          i) \(\tan 315^{\circ}\)

                $$\begin{align*} \tan 315^{\circ} &= -\tan 45^{\circ} \\ &= -1 \end{align*}$$
              ?

          j) \(\sin \frac{3\pi}{4}\)

                $$\begin{align*} \sin \frac{3\pi}{4} &= \sin \frac{\pi}{4} \\ &= \frac{\sqrt{2}}{2} \end{align*}$$
              ?

          k) \(\sin \frac{2\pi}{3}\)

                $$\begin{align*} \sin \frac{2\pi}{3} &= \sin \frac{\pi}{3} \\ &= \frac{\sqrt{3}}{2} \end{align*}$$
              ?

          l) \(\sin \frac{11\pi}{6}\)

                $$\begin{align*} \sin \frac{11\pi}{6} &= -\sin \frac{\pi}{6} \\ &= -\frac{1}{2} \end{align*}$$
              ?

          m) \(\cos \frac{7\pi}{4}\)

                $$\begin{align*} \cos \frac{7\pi}{4} &= \cos \frac{\pi}{4} \\ &= \frac{\sqrt{2}}{2} \end{align*}$$
              ?

          n) \(\cos \frac{7\pi}{6}\)

                $$\begin{align*} \cos \frac{7\pi}{6} &= -\cos \frac{\pi}{6} \\ &= -\frac{\sqrt{3}}{2} \end{align*}$$
              ?

          o) \(\cos \frac{2\pi}{3}\)

                $$\begin{align*} \cos \frac{2\pi}{3} &= -\cos \frac{\pi}{3} \\ &= -\frac{1}{2} \end{align*}$$
              ?

          p) \(\tan \frac{4\pi}{3}\)

                $$\begin{align*} \tan \frac{4\pi}{3} &= \tan \frac{\pi}{3} \\ &= \sqrt{3} \end{align*}$$
              ?

          q) \(\tan \frac{13\pi}{4}\)

                $$\begin{align*} \tan \frac{13\pi}{4} &= \tan \frac{\pi}{4} \\ &= 1 \end{align*}$$
              ?

          r) \(\tan \frac{5\pi}{6}\)

                $$\begin{align*} \tan \frac{5\pi}{6} &= -\tan \frac{\pi}{6} \\ &= -\frac{\sqrt{3}}{3} \end{align*}$$
              ?

Your Turn

Without using a calculator, evaluate$$8\sin\left(\frac{\pi}{3}\right)\cos\left(\frac{5\pi}{6}\right)$$

Your Turn

Find all angles $0\le\theta\le 2\pi$ with a cosine of $\frac{1}{2}$.

Key Facts

Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.

Taking it Deeper

Conceptual Questions to Consider

Describe how you can use the unit circle and the two special triangles to work with the exact values.

Explain how the values of $\sin \frac{\pi}{3}$ and $\cos \frac{\pi}{6}$ are related, based on the unit circle.

Common Mistakes / Misconceptions

The biggest mistake is working out the correct exact value, but then forgetting to use the unit circle to determine the sign of the answer.

Another common mistake is to mix up the sine and cosine exact values. They are opposites when in order, sine from smallest to largest.

Connecting This to Other Skills

We are using the ideas from the Unit Circle and Periodicity (3.5) in this skill.

When we start Solving Simple Equations (3.8) and Solving Equations (3.16) we will often do this in non-calculator papers where knowledge of the exact values is crucial.

We will see in working with Compound Angle Formulae (3.10) that exact values can be helpful to determine other values of the trigonometric functions.

Self-Reflection

What was the most challenging part of this skill for you?

What are you still unsure about that you need to review?

3-6 Exact Values (SL)

Required Prior Knowledge

Get Ready

Examples and Your Turns

Key Facts

Taking it Deeper